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DoITPoMS Teaching & Learning Packages Deformation of Honeycombs and Foams Derivation of Young modulus in highly porous foam

# Derivation of Young modulus in highly porous foam

Here we estimate the Young modulus of an open-cell foam, represented schematically as shown below, where each cubic cell has a side length l and is made of struts with a square cross-section of thickness t, whose Young modulus is ES. The derivation is similar to that used for the honeycomb. However because the cell is cubic rather than hexagonal, θ = 0, giving the deflection of a single half-beam, δ, as

$\delta = \frac{1}{{24}}\frac{{F{l^3}}}{{{E_{\rm{S}}}I}}$

where I is the second moment of area, where I = t4/12. This gives the deflection in the loading direction, Δx, as

$\Delta x = 2\delta = \frac{F}{{{E_{\rm{S}}}t}}{\rm{ }}{\left( {\frac{l}{t}} \right)^3}$

This gives the strain, ε, as

$\varepsilon = \frac{F}{{{E_{\rm{S}}}}}{\rm{ }} \cdot \frac{{{l^2}}}{{{t^4}}}$

The stress, σ, arising from the applied force F is

$\sigma = \frac{F}{{{l^2}}}$

This gives an expression for the Young modulus of the open-cell foam, E

$E = {E_{\rm{S}}}{\rm{ }}{\left( {\frac{t}{l}} \right)^4}$

Now

$\frac{\rho }{{{\rho _{\rm{S}}}}} \propto {\left( {\frac{t}{l}} \right)^2}$

The relative elastic modulus of the open-cell foam, E/ES, is therefore related to the relative density, ρ/ρS, according to

$\frac{E}{{{E_{\rm{S}}}}} = k{\rm{ }}{\left( {\frac{\rho }{{{\rho _{\rm{S}}}}}} \right)^2}$

where k is a constant approximately equal to 1.